In 1993, Rosenau and Hyman [1] introduced a genuinely nonlinear dispersive equation, namely, a particular generalization of the KdV equation:

$ \begin{equation*} u_{t}+(u^{m})_{x}+(u^{n})_{xxx}=0, m>0, 1<n\leq3, \end{equation*} $

which is called the $K(m, n)$ equation. They studied four cases $m, n=2, 3$ with the compacton solutions. Few years later, Ismail and Taha [2] implemented a finite difference method and a finite element method to study equations $K(2, 2)$ and $K(3, 3)$. Ismail [3] made an extension based on [2] and obtained numerical solutions of the $K(2, 3)$ equation. Frutos and Lopez-Marcos [4] presented a finite difference method for the numerical integration of the $K(2, 2)$ equation. Zhou and Tian [5] considered soliton solution of the $K(2, 2)$ equation. Xu and Tian [6] studied the peaked wave solutions of the $K(2, 2)$ equation. Zhou et al. [7] investigated kink-like and antikink-like wave solutions of the $K(2, 2)$ equation. He and Meng [8] obtained new exact explicit peakon and smooth periodic wave solutions of the $K(3, 2)$ equation. In this paper, we will discuss the osmosis $K(2, 3)$ equation

$ \begin{equation}\label{2.1.1} u_{t}+(u^{2})_{x}-(u^{3})_{xxx}=0, \end{equation} $

(1.1)

where the positive convection term denotes the convection moves and the negative dispersive term means the contracting dispersion.

The paper is organized as follows. In section 2, we discuss the bifurcation of traveling wave system related with equation (1.1) and draw its phase portraits by maple programs in Figure 1. Section 3 is devoted to obtaining exact solutions including soliton, periodic cusp wave and kink-like and antikind-like waves which belong to bounded traveling waves. A short conclusion is given in section 4.

Consider the osmosis $K(2, 3)$ equation (1.1), and change it into the following form

$ \begin{equation}\label{2.1} u_{t}+2uu_{x}-6(u_{x})^{3}-18uu_{x}u_{xx}-3u^{2}u_{xxx}=0. \end{equation} $

(2.1)

Let $u=\varphi(\xi)=\varphi(x-ct)$ be the solution for equation (1.1) or (2.1), where $c(\neq0)$ is the wave speed. Substituting $u=\varphi(x-ct)$ into (2.1) gives an ODE

$ \begin{equation}\label{2.2} -c\varphi'+2\varphi\varphi'-6(\varphi')^{3}-18\varphi\varphi'\varphi''-3\varphi^{2}\varphi'''=0. \end{equation} $

(2.2)

Integrating (2.2) once with respect to $\xi$, we have

$ \begin{equation}\label{2.3} -c\varphi+\varphi^{2}-6\varphi(\varphi')^{2}-3\varphi^{2}\varphi''=g, \end{equation} $

(2.3)

which can be rewritten as

$ \begin{equation}\label{2.4} \begin{cases} \frac{d\varphi}{d\xi}=y, \\ \frac{dy}{d\xi}=\frac{g+c\varphi-\varphi^{2}+6\varphi y^{2}}{-3\varphi^{2}}, \end{cases} \end{equation} $

(2.4)

where $g$ is an integral constant. Since (2.4) is a singular traveling wave system with a singular line $\varphi=0$, we introduce a rescaling of the time $d\xi=-3\varphi^{2}d\tau$ to transform it into a regular system. Namely,

$ \begin{equation}\label{2.5} \begin{cases} \frac{d\varphi}{d\tau}=-3\varphi^{2}y, \\ \frac{dy}{d\tau}=g+c\varphi-\varphi^{2}+6\varphi y^{2}, \end{cases} \end{equation} $

(2.5)

with a first integral

$ \begin{equation}\label{2.6} H(\varphi, y)=\varphi^{3}(\frac{1}{3}g+\frac{c}{4}\varphi-\frac{1}{5}\varphi^{2}+\frac{3}{2}\varphi y^{2}). \end{equation} $

(2.6)

Obviously, the singular traveling wave system (2.4) has the same first integral $H(\varphi, y)$ as (2.5), and accordingly both (2.4) and (2.5) have the same topological phase portraits except for the straight line $\varphi=0$. Thus, in order to understand the bifurcation sets of system (2.4), we need to study phase portraits of system (2.5). Next, we discuss the isolated and singular points of system (2.5) for their phase portraits. Let $A_{\pm}=(\frac{c\pm\sqrt{c^{2}+4g}}{2}, 0)$, then

$ \begin{equation*} \begin{array}{ll} H(A_{+})=\frac{1}{960}(c+\sqrt{c^2+4g})^{3}(16g+3c^{2}+3c\sqrt{c^2+4g}), \\ H(A_{-})=\frac{1}{960}(-c+\sqrt{c^2+4g})^{3}(-16g-3c^{2}+3c\sqrt{c^2+4g}). \end{array} \end{equation*} $

Using the qualitative analysis methods of planar dynamical system, the distribution and property of singular points are given

(1) When $g>0$, system (2.5) has a saddle point $A_{+}$ and a center $A_{-}$.

(2) When $g=0$, system (2.5) has a center $(c, 0)$.

(3) When $-\frac{c^{2}}{4}<g<0$, system (2.5) has a saddle point $A_{+}$ and a center $A_{-}$.

(4) When $g=-\frac{c^{2}}{4}$, system (2.5) has a cusp $(\frac{c}{2}, 0)$.

(5) When $g<-\frac{c^{2}}{4}$, system (2.5) has no singular point.

According to the property above, we show phase portraits of system (2.5) in Figure 1. It is easy to see a periodic cusp wave from Figure 1(1-4), a kink-like wave and a antikink-like wave from Figure 1(1-6) and soliton from Figure 1(1-7) and (1-8). Their expressions are presented in next section.

Theorem3.1  When $g=0$ and $c<0$, equation (1.1) has a periodic cusp wave solution of peak type $u=\varphi_{1}(\xi+2nT), n=0, \pm1, \pm2\cdots$ and $\xi\in[(2n-1)T, (2n+1)T]$, where $\varphi_{1}(\xi)=\frac{1}{30}\xi^{2}+\frac{5c}{4}, \ \xi\in[-T, T], $ with $T=\frac{5}{2}\sqrt{-6c}$.

Proof  In Figure 1(1-4), the periodic orbit of system (2.4) intersects $\mathit{y}$-axis, which can be expressed as $H(\varphi, y)=\varphi^{4}(\frac{c}{4}-\frac{1}{5}\varphi+\frac{3}{2}y^{2})=0$. Namely,

$ \begin{equation}\label{3.1} y=\pm\sqrt{\frac{2\varphi}{15}-\frac{c}{6}}, \quad\frac{5}{4}c\leq\varphi\leq0, \, c<0, \end{equation} $

(3.1)

which intersects $\mathit{y}$-axis at $(0, \sqrt{-\frac{c}{6}})$ and $(0, -\sqrt{-\frac{c}{6}})$, and $\mathit{x}$-axis at $(\frac{5}{4}c, 0)$. Substituting (3.1) into the system (2.4), we get

$ \begin{equation*} \begin{cases} \frac{d\varphi}{d\xi}=y, \\ \frac{dy}{d\xi}=\frac{1}{15}, \end{cases} \end{equation*} $

which gives $u=\varphi_{1}(\xi+2nT), n=0, \pm1, \pm2\cdots$ and $\varphi_{1}(\xi)=\frac{1}{30}\xi^{2}+\frac{5c}{4}, \ \xi\in[-T, T]$. In addition, $\varphi_{1}(T)=0$ reveals $T=\frac{5}{2}\sqrt{-6c}$. The proof is completed.

The graph of relevant periodic cusp wave of peak type for equation (1.1) is shown in Figure 2.

Theorem3.2  When $g=-\frac{15}{64}c^{2}$ and $c>0$, equation (1.1) has a kink-like wave solution $u=\varphi_{2}(\xi)$ and a antikink-like wave solution $u=\varphi_{3}(\xi)$,

$ \begin{equation}\label{3.2} \beta(\varphi_{2})=\beta(a)+\xi, \quad\xi\in(-\xi_{0}, +\infty), \end{equation} $

(3.2)

$ \begin{equation}\label{3.3} \beta(\varphi_{3})=\beta(a)-\xi, \quad\xi\in(-\infty, \xi_{0}), \end{equation} $

(3.3)

where $\frac{c}{2}<\varphi<\frac{5}{8}c$ and $\beta(\varphi)=\frac{\sqrt{30}}{2}[-2\sqrt{\varphi}+\frac{\sqrt{10c}}{2}\text{arctanh}\frac{2\sqrt{10\varphi}}{5\sqrt{c}}]$, $\xi_{0}=\beta(a)-\beta(\frac{c}{2})$ with $a$ is a constant satisfying $\varphi_{2}(0)=\varphi_{3}(0)=a$.

Proof  In figure 1(1-6), system (2.4) has four orbits connecting with a saddle point $A_{+}$. $L_{2}$ denotes a orbit lying on the upper-left side of $A_{+}$, and $L_{3}$ on the lower-left. Note that $H(A_{+})=0$, then $L_{2}$ and $L_{3}$ can be respectively described as the following forms for $0\leq\varphi\leq\frac{5}{8}c$:

$ \begin{equation}\label{3.4} L_{2}: y=\frac{\sqrt{2}(5c-8\varphi)}{8\sqrt{15\varphi}}, \end{equation} $

(3.4)

$ \begin{equation}\label{3.5} L_{3}: y=-\frac{\sqrt{2}(5c-8\varphi)}{8\sqrt{15\varphi}}. \end{equation} $

(3.5)

Let $\varphi_{2}(\xi)$ and $\varphi_{3}(\xi)$ be respective solutions of system (2.4) on $L_{2}$ and $L_{3}$. Substituting (3.4) and (3.5) into the first equation of system (2.4) separately and integrating along orbits $L_{2}, L_{3}$, respectively, we get

$ \begin{equation}\label{3.6} \int_{\varphi_{2}}^{a}\frac{8\sqrt{15s}}{\sqrt{2}(5c-8s)}ds=\int_{\xi}^{0}ds, \end{equation} $

(3.6)

$ \begin{equation}\label{3.7} \int_{a}^{\varphi_{3}}\frac{-8\sqrt{15s}}{\sqrt{2}(5c-8s)}ds=\int_{0}^{\xi}ds, \end{equation} $

(3.7)

which imply implicit functions in (3.2) and (3.3) by maple program.

Besides, assume that $\varphi_{2}(\xi)\rightarrow\frac{c}{2}$ as $\xi\rightarrow-\xi_{0}$, and $\varphi_{3}(\xi)\rightarrow\frac{c}{2}$ as $\xi\rightarrow\xi_{1}$, then we have from (3.6)-(3.7) that

$ \begin{equation*} \xi_{0}=\xi_{1}=\int_{a}^{\frac{c}{2}}\frac{-8\sqrt{15s}}{\sqrt{2}(5c-8s)}ds=\beta(a)-\beta(\frac{c}{2}). \end{equation*} $

The proof is completed.

The graph of relevant kink-like and antikink-like waves for equation (1.1) is shown in Figure 3.

Theorem3.3  When $-\frac{c^2}{4}<g<-\frac{15}{64}c^{2}$ and $c>0$, equation (1.1) has a soliton solution

$ \begin{equation}\label{3.8} \begin{cases} u(x, t)=\varphi(x-ct), \\ \phi=\arccos(\frac{A+l_{1}-\varphi}{A-l_{1}+\varphi}), \\ 2hAE(\phi, k)-h(l+l_{1}-A-\frac{l^{2}}{l-l_{1}+A})F(\phi, k)-\frac{2Al^{2}h}{(\alpha^{2}-1)(l-l_{1}-A)^{2}}\Pi(\phi, \frac{\alpha^{2}}{\alpha^{2}-1}, k)\\ -2hA\frac{\sqrt{1-k^{2}\phi^{2}}\phi}{1+\sqrt{1-\phi^{2}}}\\+\frac{Al^{2}h\alpha\sqrt{\alpha^{2}-1}}{(l-l_{1}-A)^{2}(\alpha^{2}-1)\sqrt{k^{2}+\alpha^{2} -k^{2}\alpha^{2}}}\ln\frac{\sqrt{k^{2}+\alpha^{2}-k^{2}\alpha^{2}}\phi+\sqrt{(\alpha^{2}-1)(1-k^{2}\phi^{2})}}{\sqrt{k^{2}+\alpha^{2}-k^{2}\alpha^{2}}\phi -\sqrt{(\alpha^{2}-1)(1-k^{2}\phi^{2})}}=\sqrt{\frac{2}{15}}|\xi|, \end{cases} \end{equation} $

(3.8)

where $l=\frac{1}{2}(c+\sqrt{c^{2}+4g}), \, b_{1}=-\frac{1}{2}l_{2}, \ a_{1}^{2}=l_{3}-\frac{1}{4}l_{2}^{2}, \, A=\sqrt{(b_{1}-l_{1})^{2}+a_{1}^{2}}, \, k=\sqrt{\frac{A+b_{1}-l_{1}}{2A}}, \, h=\frac{1}{\sqrt{A}}, \, \alpha=\frac{l-l_{1}+A}{l-l_{1}-A}$ with $l_{1}, l_{2}, l_{3}$ given in (3.12). Here, $F(\cdot, \cdot)$ is the normal elliptic integral of the first kind, $E(\cdot, \cdot)$ is the normal elliptic integral of the second kind and $\Pi(\cdot, \cdot, \cdot)$ is the Legendre's incomplete elliptic integral of the third kind (See [9]).

Proof  In figure 1(1-7), system (2.4) has a homoclinic loop consisting of a saddle point $A_{+}$, which can be expressed as $H(\varphi, y)=\varphi^{3}(\frac{1}{3}g+\frac{c}{4}\varphi-\frac{1}{5}\varphi^{2}+\frac{3}{2}\varphi y^{2})=H(A_{+})$. Namely,

$ \begin{equation}\label{3.9} \frac{3}{2}\varphi^{4}y^{2}=\frac{1}{5}(\varphi-\frac{c+\sqrt{c^{2}+4g}}{2})^{2}(\varphi^{3}+m_{1}\varphi^{2}+m_{2}\varphi+m_{3}), \end{equation} $

(3.9)

where $m_{1}=-\frac{c}{4}+\sqrt{c^{2}+4g}, \, m_{2}=\frac{c^{2}}{4}+\frac{c\sqrt{c^{2}+4g}}{4}+\frac{4g}{3}, \, m_{3}=\frac{c^{3}}{8}+ \frac{c^{2}\sqrt{c^{2}+4g}}{8}+\frac{7cg}{12}+\frac{g\sqrt{c^{2}+4g}}{3}$. Let

$ \begin{equation}\label{3.10} f(\varphi)=\varphi^{3}+m_{1}\varphi^{2}+m_{2}\varphi+m_{3}. \end{equation} $

(3.10)

By introducing a new variable, $x=\varphi+\frac{m_{1}}{3}$, (3.10) becomes

$ \begin{equation*} f(\varphi)=x^{3}+px+q, \end{equation*} $

where $p=-\frac{5c^{2}}{48}+\frac{5c\sqrt{c^{2}+4g}}{12}, \, q=\frac{5c^{3}}{864}+\frac{65c^{2}\sqrt{c^{2}+4g}}{432}+\frac{5cg}{36}+\frac{5g\sqrt{c^{2}+4g}}{27}$. Besides, $(\frac{q}{2})^{2}+(\frac{p}{3})^{3}>0$ under $-\frac{c^2}{4}<g<-\frac{15}{64}c^{2}$ and $c>0$, then (3.10) has a sole real root. Thus, (3.9) becomes

$ \begin{equation}\label{3.11} y=\sqrt{\frac{2}{15}}\frac{(l-\varphi)\sqrt{(\varphi-l_{1})(\varphi^{2}+l_{2}\varphi+l_{3})}}{\varphi^{2}}, \, l_{1}<\varphi<l, \end{equation} $

(3.11)

where

$ \begin{equation*} \begin{array}{ll} l_{1}=-\frac{1}{12}\bigg\{-\Big[-5c^{3}-130c^{2}\sqrt{c^{2}+\\ 4g}-120cg-160\sqrt{c^{2}+4g}g\\ +20\sqrt{(3c^{2}+2g)(9c^{4}+9c^{3}\sqrt{c^{2}+4g}\\ +66c^{2}g+48c\sqrt{c^{2}+4g}g+128g^{2})}~\Big]^{\frac{2}{3}}-5c^{2}\\ +20c\sqrt{c^{2}+4g}-c\Big[-5c^{3}-130c^{2}\sqrt{c^{2}+4g}-120cg -160\sqrt{c^{2}+4g}g\\ +20\sqrt{(3c^{2}+2g)(9c^{4}+9c^{3}\sqrt{c^{2}+4g}+\\ 66c^{2}g+48c\sqrt{c^{2}+4g}g+128g^{2})}~\Big]^{\frac{1}{3}}\\ +4\sqrt{c^{2}+4g}\Big[-5c^{3}-130c^{2}\sqrt{c^{2}+4g}-120cg-160\sqrt{c^{2}+4g}g\\ +20\sqrt{(3c^{2}+2g)(9c^{4}+9c^{3}\sqrt{c^{2}+4g}\\ +66c^{2}g+48c\sqrt{c^{2}+4g}g+128g^{2})}\Big]^{\frac{1}{3}}\bigg\}/\Big[-5c^{3}\\ -130c^{2}\sqrt{c^{2}+4g}-120cg-160\sqrt{c^{2}+4g}g\\ +20\sqrt{(3c^{2}+2g)(9c^{4}+9c^{3}\sqrt{c^{2}+4g}+\\ 66c^{2}g+48c\sqrt{c^{2}+4g}g+128g^{2})}~\Big]^{\frac{1}{3}}, \\ l_{2}=-\frac{1}{60c^{2}}\bigg\{2250c^{7}-9000c^{6}\sqrt{c^{2}+4g}+\\ 19200c^{5}g-76800c^{4}g\sqrt{c^{2}+4g}+40960c^{3}g^{2}\\ -163840c^{2}g^{2}\sqrt{c^{2}+4g} -(1125c^{6}+9600c^{4}g+20480c^{2}g^{2})\\ \Big[-5c^{3}-130c^{2}\sqrt{c^{2}+4g}-120cg-160\sqrt{c^{2}+4g}g+20\\ \sqrt{162c^{3}\sqrt{c^{2}+4g}g+216c^{4}g+\\ 516c^{2}g^{2}+96cg^{2}\sqrt{c^{2}+4g}+27c^{5}\sqrt{c^{2}+4g} +27c^{6}+256g^{3}}\Big]^{\frac{1}{3}}\\ +\Big[-5c^{3}-130c^{2}\sqrt{c^{2}+4g}-120cg-160\sqrt{c^{2}+4g}g+20\\ \sqrt{162c^{3}\sqrt{c^{2}+4g}g+216c^{4}g+\\ 516c^{2}g^{2}+96cg^{2}\sqrt{c^{2}+4g}+27c^{5}\sqrt{c^{2}+4g} +27c^{6}+256g^{3}}\Big]^{\frac{2}{3}}\\ \Big[+225c^{5}+450c^{4}\sqrt{c^{2}+4g}+\\ 1560c^{3}g+2400c^{2}\sqrt{c^{2}+4g}g+2560cg^{2} +2048\sqrt{c^{2}+4g}g^{2}\\ +(68c^{2}+32c\sqrt{c^{2}+4g}+256g)\\ \sqrt{162c^{3}\sqrt{c^{2}+4g}g +216c^{4}g+516c^{2}g^{2}+96cg^{2}\sqrt{c^{2}+4g}+\\ 27c^{5}\sqrt{c^{2}+4g}+27c^{6}+256g^{3}}\Big]\bigg\}/\\ (15c^{2}+64g)^{2}, \\ l_{3}=\frac{1}{360c^{2}}\bigg\{450c^{6}+450c^{5}\sqrt{c^{2}+\\ 4g}+4320c^{4}g+1920c^{3}\sqrt{c^{2}+4g}g+10240c^{2}g^{2}+\\ \Big[-5c^{3}-130c^{2}\sqrt{c^{2}+4g}-120cg-160\sqrt{c^{2}+4g}g+20\\ \sqrt{162c^{3}\sqrt{c^{2}+4g}g+216c^{4}g+516c^{2}g^{2}\\ +96cg^{2}\sqrt{c^{2}+4g}+27c^{5}\sqrt{c^{2}+4g}+27c^{6}+256g^{3}}\Big]^{\frac{1}{3}}\\ \cdot\Big[225c^{5}+225c^{4}+1260c^{3}g +960c^{2}g\sqrt{c^{2}+4g}+1280cg^{2}\\ +(10c^{2}+40c\sqrt{c^{2}+4g})\\ \sqrt{162c^{3}\sqrt{c^{2}+4g}g+216c^{4}g+516c^{2}g^{2}+96cg^{2}\sqrt{c^{2}+4g}\\ +27c^{5}\sqrt{c^{2}+4g}+27c^{6}+256g^{3}}\Big]\\ +\Big[-5c^{3}-130c^{2}\sqrt{c^{2}+4g}-120cg-160\sqrt{c^{2}+4g}g+20\\ \sqrt{162c^{3}\sqrt{c^{2}+4g}g+216c^{4}g+516c^{2}g^{2}\\ +96cg^{2}\sqrt{c^{2}+4g}+27c^{5}\sqrt{c^{2}+4g}+27c^{6}+256g^{3}}\Big]^{\frac{2}{3}}\\ \cdot\Big[-15c^{4} +225c^{4}-15c^{3}\sqrt{c^{2}+4g}-124c^{2}g-64cg\sqrt{c^{2}+4g}\\ -(2c+8\sqrt{c^{2}+4g})\\ \quad\sqrt{162c^{3}\sqrt{c^{2}+4g}g+216c^{4}g+516c^{2}g^{2}+96cg^{2}\sqrt{c^{2}+4g}\\ +27c^{5}\sqrt{c^{2}+4g}+27c^{6}+256g^{3}}\\ \quad-256g^{2}\Big]\bigg\}/(15c^{2}+64g), \end{array} \end{equation*} $

(3.12)

with $l^{2}_{2}-4l_{3}>0$. Substituting (3.11) into the first equation of system (2.4) and integrating along the homoclinic orbit, we get

$ \begin{equation} \begin{array}{ll} -lhF(\phi, k)-h(l_{1}-A)F(\phi, k)-2hA\int_{0}^{u_{1}}\frac{du}{1+cn\, u}+l^{2}h\int_{0}^{u_{1}}\frac{1+cn\, udu}{(l-l_{1}+A)cn\, u+l-l_{1}-A]}\\ ~=\sqrt{\frac{2}{15}}|\xi|, \end{array} \end{equation} $

(3.13)

where $cn\, u_{1}=\cos\phi, cn\, u=cn(u, k)$ is Jacobian elliptic function (see [9]). Therefore, (3.8) holds by computation.

The graph of relevant soliton for equation (1.1) is shown in Figure 4(4-1).

Remark  When $-\frac{c^2}{4}<g<0$ and $c<0$ (Figure 1(1-8)), equation (1.1) also has a soliton solution and (3.8) holds. Its graph is shown in Figure 4(4-2).

In this paper, using the qualitative analysis methods of planar dynamical system, we studied the bifurcation of the osmosis $K(2, 3)$ equation and obtained exact periodic cusp wave solution, kink-like and antikink-like wave solutions, and soliton solution.